About 3,700 results
|Quadratic Transformations The formulas 3.8(1—24) may be viewed as a |
transformation of the 2F1 in the variable, actually a linear ... The basic formulas
are due to Gauss and Kummer and a complete list is due to Goursat; see Erdélyi
|1662 Kummer's Function where 2F1(a, b, c, z) is the HYPERGEOMETRIC |
FUNCTION with m #–1/2, -1, -3/2, ..., and T(z) is the ... Kummer's Function
CONFLUENT HYPERGEOMETRIC FUNCTION Kummer's Quadratic
Transformation A ...
|dimensional spaces of solutions of certain differential equations. In order to relate |
them to each other, we must use Kummer's quadratic transformation formula. This
involves certain substitutions which lift the correlation functions of t = 22/21 to ...
|8.18 Show that 2/ r(i[c + a])r(i[c-a+ 1]) .19 Use (8.6.23) (with a and b interchanged|
) to derive Kummer's quadratic transformation (6.1.11). .20 Show that / 1 1\ r(i[a +
*+l])r(i) F[a,b,-[a + b+l]; »- 2 2/ r(I[fl+l])r(I[6+l]) 8.21 Verify the identity (8.4.5).
|Quadratic Transformations In §5.4 we mentioned Kummer's 24 solutions of the |
hypergeornetric differential equation. ... (1881) made a thorough study of another
kind of transformation, although the basic ideas are due to Gauss and Kummer.
|At the end of the second part of this paper Kummer wrote that he had tried to |
extend his results to ... The rest of the quadratic transformations come from using
linear fractional relations and connections between three solutions of the ...
|(6,202) 2 4x3 Kummer's quadratic transformation formulas for the hypergeometric |
functions give (see , 15.3.22) 1 1 Fi | –2 ji , –2 j2; – — ji – j2; —— 2 ( J1 J25 2
J1 - J2 :) 1 1 \* = 2 Fl (- - - - -(- #) ) • (6.203) 2 2x3 Now for any 6 Ji ..J2 ji y y - • y ...
|Linear, Quadratic and Cubic Transformations Pfajf-Kummer transformations: 2F1|
<a, b; C; 1) = (1—Z)—a2F1 (4, c-b; C; L) <19) z—1 (c<,-:25; |arg(1—z)| §7T—€ (0
<€ <rr)); 2F1<a, b; C; 1) = <1- Z)_b2F1 <1», c- 4; C; <20) Z (c˘Z6; |arg(1—z)| ...
|In Pfaff ,s transformation. (2.2.6) c change x to x/b and let b -> oo to get Kummer,s |
first transformation, (4.1.11) A similar procedure applied to the quadratic
transformation, leads to Kummer,s second transformation, a •+ •" ' (4,U2) Finally,